| L(s) = 1 | + (−32 + 55.4i)2-s + (301. + 523. i)3-s + (−2.04e3 − 3.54e3i)4-s + (2.52e3 − 4.36e3i)5-s − 3.86e4·6-s + (3.02e5 + 7.52e4i)7-s + 2.62e5·8-s + (6.14e5 − 1.06e6i)9-s + (1.61e5 + 2.79e5i)10-s + (5.42e5 + 9.38e5i)11-s + (1.23e6 − 2.14e6i)12-s + 4.14e6·13-s + (−1.38e7 + 1.43e7i)14-s + 3.04e6·15-s + (−8.38e6 + 1.45e7i)16-s + (7.24e7 + 1.25e8i)17-s + ⋯ |
| L(s) = 1 | + (−0.353 + 0.612i)2-s + (0.239 + 0.414i)3-s + (−0.249 − 0.433i)4-s + (0.0721 − 0.125i)5-s − 0.338·6-s + (0.970 + 0.241i)7-s + 0.353·8-s + (0.385 − 0.667i)9-s + (0.0510 + 0.0884i)10-s + (0.0922 + 0.159i)11-s + (0.119 − 0.207i)12-s + 0.238·13-s + (−0.491 + 0.508i)14-s + 0.0690·15-s + (−0.125 + 0.216i)16-s + (0.728 + 1.26i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 14 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.179 - 0.983i)\, \overline{\Lambda}(14-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14 ^{s/2} \, \Gamma_{\C}(s+13/2) \, L(s)\cr =\mathstrut & (0.179 - 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(7)\) |
\(\approx\) |
\(1.43033 + 1.19258i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.43033 + 1.19258i\) |
| \(L(\frac{15}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (32 - 55.4i)T \) |
| 7 | \( 1 + (-3.02e5 - 7.52e4i)T \) |
| good | 3 | \( 1 + (-301. - 523. i)T + (-7.97e5 + 1.38e6i)T^{2} \) |
| 5 | \( 1 + (-2.52e3 + 4.36e3i)T + (-6.10e8 - 1.05e9i)T^{2} \) |
| 11 | \( 1 + (-5.42e5 - 9.38e5i)T + (-1.72e13 + 2.98e13i)T^{2} \) |
| 13 | \( 1 - 4.14e6T + 3.02e14T^{2} \) |
| 17 | \( 1 + (-7.24e7 - 1.25e8i)T + (-4.95e15 + 8.57e15i)T^{2} \) |
| 19 | \( 1 + (1.39e8 - 2.41e8i)T + (-2.10e16 - 3.64e16i)T^{2} \) |
| 23 | \( 1 + (4.18e7 - 7.25e7i)T + (-2.52e17 - 4.36e17i)T^{2} \) |
| 29 | \( 1 - 6.32e8T + 1.02e19T^{2} \) |
| 31 | \( 1 + (1.65e9 + 2.87e9i)T + (-1.22e19 + 2.11e19i)T^{2} \) |
| 37 | \( 1 + (-8.76e9 + 1.51e10i)T + (-1.21e20 - 2.10e20i)T^{2} \) |
| 41 | \( 1 - 1.15e10T + 9.25e20T^{2} \) |
| 43 | \( 1 - 4.19e10T + 1.71e21T^{2} \) |
| 47 | \( 1 + (2.47e10 - 4.28e10i)T + (-2.73e21 - 4.72e21i)T^{2} \) |
| 53 | \( 1 + (-1.09e11 - 1.89e11i)T + (-1.30e22 + 2.25e22i)T^{2} \) |
| 59 | \( 1 + (6.67e10 + 1.15e11i)T + (-5.24e22 + 9.09e22i)T^{2} \) |
| 61 | \( 1 + (-6.84e10 + 1.18e11i)T + (-8.09e22 - 1.40e23i)T^{2} \) |
| 67 | \( 1 + (4.61e11 + 7.99e11i)T + (-2.74e23 + 4.74e23i)T^{2} \) |
| 71 | \( 1 + 9.00e11T + 1.16e24T^{2} \) |
| 73 | \( 1 + (9.23e11 + 1.59e12i)T + (-8.35e23 + 1.44e24i)T^{2} \) |
| 79 | \( 1 + (1.46e12 - 2.53e12i)T + (-2.33e24 - 4.04e24i)T^{2} \) |
| 83 | \( 1 - 4.12e12T + 8.87e24T^{2} \) |
| 89 | \( 1 + (2.20e12 - 3.81e12i)T + (-1.09e25 - 1.90e25i)T^{2} \) |
| 97 | \( 1 - 3.35e12T + 6.73e25T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−16.67166743141026916463618247198, −15.16588297110151605472091204005, −14.47157762067657935910685117069, −12.50745981187785423019486515767, −10.62535120057788951332036743896, −9.146455061108174495024850644256, −7.84172229070845956759314329917, −5.89733127094827693787108809787, −4.10061134584600035771671004563, −1.41975547149108629559314415842,
0.994437450618950455823993121946, 2.51517502295507405607491982020, 4.70834082828787489641643198600, 7.27292255451508168381304373020, 8.596820644764143442176110871480, 10.40760862771394139425117366906, 11.63742688887621372956834147745, 13.23336999376398239431629945865, 14.36003440123387689860160047222, 16.28231351849401552132664839883
